I have noticed in both the fighting, but far more so with the shooting sequences, the game seems to speed up substantially fast, to the point that it is hard to be effective. I am missing something here?

Just want to say this is happening to me too. Let us know if a solution is coming, GOG.

Also, swimming doesn't seem to work.

Okay so I have things working, for starters, I noticed this when I confronted a guy at a bar and was immediately killed. Turns out my health was super low and I could not even take one hit. once you have health, this issue is just that things seem sped up. I have been able to compensate thus far, but it can be tricky.

As far as swimming, I wasn't using the right control button which pretty much makes everything else work right while swimming.

A bit of a shot in the dark, but do you have vsync forced off? That can often cause timing problems in older games that rely on sane framerates to keep their timings in check.

Note that ati drivers have slightly odd 'off unless application specifies/on unless application specifies' options which don't always work correctly on older games, where using 'off unless application specifies' can disable vsync whether the game wants it or not.